Two-band description of resonant superfluidity in atomic Fermi gases

# Two-band description of resonant superfluidity in atomic Fermi gases

Lianyi He, Hui Hu, Xia-Ji Liu 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2 Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria, 3122, Australia
July 14, 2019
###### Abstract

Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described by the atom-molecule theory, which treats the closed channel as a noninteracting point boson. In this work we present a theoretical description of the resonant superfluidity in analogy to the two-band superconductors. We employ the underlying two-channel scattering model of Feshbach resonance where the closed channel is treated as a composite boson with binding energy and the resonance is triggered by the microscopic interchannel coupling . The binding energy naturally serves as an energy scale of the system, which has been sent to infinity in the atom-molecule theory. We show that the atom-molecule theory can be viewed as a leading-order low-energy effective theory of the underlying fermionic theory in the limit and , while keeping the phenomenological atom-molecule coupling finite. The resulting two-band description of the superfluid state is in analogy to the BCS theory of two-band superconductors. In the dilute limit , the two-band description recovers precisely the atom-molecule theory. The two-band theory provides a natural approach to study the corrections because of a finite binding energy in realistic experimental systems. For broad and moderate resonances, the correction is not important for current experimental densities. However, for extremely narrow resonance, we find that the correction becomes significant. The finite binding energy correction could be important for the stability of homogeneous polarized superfluid against phase separation in imbalanced Fermi gases across a narrow Feshbach resonance.

###### pacs:
03.75.Ss, 05.30.Fk, 67.85.Lm, 74.20.Fg

## I Introduction

It is widely accepted that a crossover from the BCS superfluidity to the Bose-Einstein condensation (BEC) of molecules can be realized in an attractive Fermi gas by tuning the attraction from weak to strong BCS-BEC (). This interesting phenomenon has been experimentally observed in ultracold Fermi gases of alkali-metal atoms EXP () (such as Li and K). In these experiments, the attractive strength is effectively tuned by means of the Feshbach resonances (FRs). The basic mechanism of the FR is the coupling between different scattering channels in alkali-metal atoms in a magnetic field Feshbach (); book ().

The scattering channels of the alkali-metal atoms are characterized by the eigenstates of the single-particle hyperfine Hamiltonian in a magnetic field . The main contribution to the atom-atom interaction is the electrostatic central potential which also induces the couplings among different scattering channels. Because of these interchannel coupling, a FR occurs when the bound-state level of a certain closed channel coincides with the threshold of a certain open channel. A schematic plot for this mechanism is shown in Fig. 1. In the vicinity of an -wave FR, the low-energy scattering amplitude for the open channel is given by

 f(p)=1pcotδ(p)−ip, (1)

where the scattering phase shift can be well parametrized as Feshbach (); book ()

 pcotδ(p)=−1abgE−γ(B−B0)E−γ(B−B0)+γBΔ. (2)

Here is the scattering energy, with being the atom mass, is the resonance point, is the resonance width, is the difference of the magnetic moments between the two-channels, and is the background scattering length. The units will be used throughout. The magnetic detuning then tunes the effective scattering length of the open channel. Near the FR, can be expanded as

 pcotδ(p)≃−1aeff+12reffp2+⋯, (3)

with an effective scattering length

 aeff=abg(1−BΔB−B0) (4)

and a negative effective range

 reff=−2MabgγBΔ. (5)

For many-body systems with total density , we normally define a Fermi wave vector and corresponding Fermi energy . So far, most of the experimental studies focus on broad resonances, where . In this case, the many-body physics near the FR is universal and can be well described by a single-channel model. The universal many-body physics can be obtained from Monte Carlo simulations QMC () by using any short-ranged potential with the same scattering length and negligible effective range. Recently, resonantly interacting Fermi gas with a large effective range has been experimentally realized by using the narrow resonance of Li at G Expnarrow ().

For general resonances, a popular effective model is the atom-molecule model atom-molecule01 (); atom-molecule02 (); atom-molecule03 (); atom-molecule04 (), which precisely reproduces the low-energy scattering amplitude parametrized by (2). The model Hamiltonian can be written as

 H=Hf+Hb+Hfb, (6)

where the atom part,

 Hf=∑σ=↑,↓∫d3rψ†σ(−∇22M)ψσ+u0∫d3rψ†↑ψ†↓ψ↓ψ↑, (7)

the molecule part,

 Hb=∫d3rϕ†m(−∇24M+δ0)ϕm, (8)

and the atom-molecule coupling part,

 Hbf=g0∫d3r(ϕ†mψ↓ψ↑+ϕmψ†↑ψ†↓). (9)

Here denotes the open-channel fermions and denotes the closed-channel molecules. The couplings and and the detuning are bare quantities. They should be renormalized by using the physical background scattering length , resonance width , and detuning . In this model, the closed channel is treated as a point boson and the FR is triggered by the atom-molecule coupling .

Another idea to study the narrow resonance is to use a well plus barrier potential barrier () which can reproduce a large and negative effective range. However, it is essentially a single-channel model which lacks the information of the closed channel. Actually, it has been shown that the closed channel dominates in the narrow resonance limit atom-molecule04 (). In this paper, we go back to the underlying two-channel Hamiltonian which treats both the open and the closed channels as fermions Bruun (). We show that the (renormalized) atom-molecule coupling is related to the underlying inter-channel coupling and the closed-channel binding energy through

 g=U12√(Mε0)3/22π. (10)

The binding energy of the closed-channel bound state, which serves as a natural energy scale of the system, is automatically sent to infinity in the atom-molecule model. We show explicitly that the atom-molecule model can be viewed as a low-energy effective theory of the underlying two-channel theory in the limit and , while keeping finite. For many-body physics, the resonant Fermi gas can be viewed as a two-band superfluid with a large band offset . Therefore, the underlying two-channel Hamiltonian will be referred to as a two-band model in this paper. In the dilute limit , the prediction of the many-body physics becomes essentially the same as the atom-molecule model. However, in realistic experimental systems, the ratio is small but finite. For broad and moderate resonances, the correction due to nonvanishing is not important. However, for extremely narrow resonance, this correction becomes significant.

The paper is organized as follows. In Sec. II we briefly review the atom-molecule model description of resonant superfluidity. In Sec. III we calculate the low energy scattering amplitude in a two-band model and show that the atom-molecule model can be viewed as a low-energy effective theory. We formulate the resonant Fermi gas as a two-band superfluid in Sec. IV and study its dilute limit in Sec. V. We apply the two-band description to study the narrow resonance of Li in Sec. VI. The paper is summarized in Sec. VII.

## Ii Review: Atom-Molecule Theory

In this section, we briefly review the atom-molecule theory of resonant superfluidity in atomic Fermi gases. We introduce the renormalization of the atom-molecule model and its description of the superfluid state atom-molecule02 (); atom-molecule03 (); atom-molecule04 ().

### ii.1 Renormalization of the model

To renormalize the model, we first calculate the two-body scattering amplitude . The Lippmann-Schwinger equation for two-fermion scattering can be expressed by using an energy-dependent interaction vertex,

 V(E)=u0+g20E−δ0. (11)

 T(E)=V(E)1−V(E)Π(E), (12)

where the two-particle bubble function is given by

 Π(E)=∑k1E+iϵ−2εk. (13)

The integral over is divergent and we introduce a cutoff . Completing the integral we obtain

 Π(E)=−MΛ2π2+M4π√−M(E+iϵ). (14)

The scattering amplitude takes the form of Eq. (1), where reads

 pcotδ(p)=−2Λπ−4πM(u0+g20E−δ0)−1. (15)

Next we match the above result to the physical result (2). The renormalizability of the model requires that the equality

 −2Λπ−4πM(u0(Λ)+g20(Λ)E−δ0(Λ))−1 (16) = −1abgE−δE−δ+γBΔ

holds for an arbitrary value of the scattering energy through proper cutoff dependence of the bare couplings and the detuning. Defining the renormalized couplings , , and detuning , we obtain

 u0(Λ)=u1−η(Λ)u, g0(Λ)=g1−η(Λ)u, δ0(Λ)=δ+g2η(Λ)1−η(Λ)u, (17)

where . When the background scattering length is neglected, i.e., , only the detuning needs renormalization. In this case, we have and .

### ii.2 Superfluid state

The partition function of the many-body system can be expressed as

 Z=∫[dψ][dψ†][dϕm][dϕ†m]exp(−Sψ,ϕ), (18)

 Sψ,ϕ = ∫dx∑σ=↑,↓ψ†σ(x)(∂τ−μ)ψσ(x) (19) + ∫dx ϕ†m(x)(∂τ−2μ)ϕm(x)+∫β0dτH.

Here , with being the imaginary time, and , with being the temperature of the system. Here we have introduced the chemical potential which is conjugate to the total particle number. To decouple the four-fermion interaction term, we introduce an auxiliary field . By performing the Hubbard-Stratonovich transformation, we obtain

 Z=∫[dφ][dφ†][dϕm][dϕ†m]exp(−Seff) (20)

 Seff = ∫dx ϕ†m(x)(∂τ−∇24M+δ0−2μ)ϕm(x) (21) − ∫dx |φ(x)|2u0−∫dx TrlnG−1[φ,ϕm],

with the inverse fermion Green’s function given by

 G−1=⎛⎜⎝−∂τ+∇22M+μφ+g0ϕmφ†+g0ϕ†m−∂τ−∇22M−μ⎞⎟⎠δ(x−x′). (22)

In the superfluid phase, the two boson fields and generate nonzero expectation values. We define

 Δb=g0⟨ϕm(x)⟩,      Δf=⟨φ(x)⟩. (23)

In the mean-field approximation, the grand potential at is given by

 Ω0=δ0−2μg20|Δb|2−|Δf|2u0+∑|k|<Λ(ξk−Ek), (24)

where and , with . The next step is to remove the cutoff dependence by using the physical quantities , , and .

To renormalize the grand potential, we note that and are cutoff dependent and therefore not physical quantities atom-molecule02 (). To show this, we use the stationary condition

 ∂Ω0∂Δ∗b=δ0−2μg20Δb−∑kΔ2Ek=0, ∂Ω0∂Δ∗f=−Δfu0−∑kΔ2Ek=0 (25)

to obtain

 Δb=g202μ−δ0Δfu0. (26)

Then we have

 Δb=g20/(2μ−δ0)u0+g20/(2μ−δ0)Δ, Δf=u0u0+g20/(2μ−δ0)Δ. (27)

Therefore, and are cutoff dependent. To renormalize the grand potential, we should regard them as dependent quantities and express the grand potential in terms of the finite quantity . Finally, we obtain

 Ω0(Δ)=−|Δ|2u0+g20/(2μ−δ0)+∑|k|<Λ(ξk−Ek). (28)

Using the fact that

 1u0+g20/(2μ−δ0)=1ueff−η(Λ), (29)

we obtain a cutoff-independent expression,

 Ω0(Δ)=−|Δ|2ueff+∑k(ξk−Ek+|Δ|22εk), (30)

where

 ueff=u+g22μ−δ. (31)

The gap equation can be derived from . We have

 1ueff=∑k(12εk−12Ek). (32)

Meanwhile, the total density is obtained through . We obtain

 n=∑k(1−ξkEk)+nm, (33)

where the contribution from the closed channel is given by

 nm=2|Δ|2g2(1−uueff)2. (34)

From the above coupled equations, we can solve the pairing gap , the chemical potential , and the closed-channel fraction at given detuning . For sufficiently large coupling , the result reproduces the universality predicted by the single-channel model. For finite temperature properties and beyond-mean-field treatment, we refer to Refs. atom-molecule02 (); atom-molecule03 (); atom-molecule04 ().

## Iii A Two-Band Model for Feshbach Resonance

The precise prediction of the FRs relies on solving the microscopic multichannel scattering problem with known microscopic interaction potentials. However, the scattering problem near a specific FR can be attributed to an effective two-channel problem. Let us consider a two-channel Hamiltonian Bruun (), where

 H0=∑n=1,2∑σ=↑,↓∫d3rψ†nσ(r)(−∇22M+εnσ)ψnσ(r). (35)

Here and correspond to the open channel and the closed channel, respectively. The interaction part is

 Hint=∑m,n=1,2∫d3r∫d3r′φ†m(r)Vmn(|r−r′|)φn(r′), (36)

where we use the notation

 φn(r)=ψn↓(r)ψn↑(r). (37)

In this second quantization form, the thresholds are put into the free part . Therefore, the interaction potential for . It includes both intra- and inter-channel interactions.

The threshold energies can be further simplified. Without loss of generality, we set

 ε1↑=ε1↓=0,      ε2↑=ε2↓=12εth. (38)

For a many-body system, the difference between and can be absorbed into the definition of the chemical potentials.

### iii.1 Low-energy scattering amplitude

The effective range of the microscopic potential introduces an energy scale

 εr=1Mr20. (39)

At low scattering energy , the shape of the microscopic interaction potential is not important. It can be safely replaced with a contact one . For many-body physics, this means that all kinds of short-ranged potential leads to the same predictions in the dilute limit Bruun (). By making use of the contact potential, the Lippmann-Schwinger equation of the scattering matrix becomes an algebra equation,

 (T11(E)T12(E)T21(E)T22(E))−1=(V11V12V21V22)−1−(B1(E)00B2(E)), (40)

where the two-particle bubble functions are given by

 Bn(E)=∑k1E+iϵ−εnth−2εk. (41)

Here and . Note that we have set and without loss of generality. The cost of the contact interaction is that the integral over the fermion momentum becomes divergent. We introduce a cutoff for and obtain

 Bn(E)=−MΛ2π2+Πn(E), (42)

where

 Π1(E)=M4π√−M(E+iϵ), Π2(E)=M4π√−M(E+iϵ−εth). (43)

The divergence can be removed by using the renormalized coupling matrix . It is related to the bare coupling matrix by Bruun ()

 (U11U12U21U22)−1=(V11V12V21V22)−1+(η(Λ)00η(Λ)). (44)

Without loss of generality, we set . Then the Lippmann-Schwinger equation becomes cutoff independent,

 (T11(E)T12(E)T21(E)T22(E))−1=(U11U12U21U22)−1−(Π1(E)00Π2(E)). (45)

Next we relate the elements of to physical observables. In general, both the coupling and the threshold energy depend on the magnetic field . However, near the FR we may safely neglect the dependence of the coupling . The threshold energy can be well parametrized as

 εth(B)=ε0+δ(B), (46)

where is the binding energy of the closed-channel molecule and is the magnetic detuning. The binding energy serves as another energy scale of the system. For an atomic system, we normally have the hierarchy . For the problem of FR, low-energy scattering means that the scattering energy . This is actually the simplest model for FR in atomic systems. If we know the explicit dependence of the microscopic interaction potential and the threshold energy , we can have better description of the dependence Bruun ().

Solving the Lippmann-Schwinger equation, we obtain the matrix for the open channel,

 T−111(E)=[U11+U212Π2(E)1−U22Π2(E)]−1−Π1(E). (47)

A FR occurring at requires that diverges at . Since , we obtain

 1U22=Π2(0)=M4π√Mε0. (48)

This equation clearly shows that the bound-state level of the closed channel coincides with the threshold of the open channel when FR occurs. The scattering amplitude for the open channel is defined as . At low scattering energy , is real and . Therefore, takes the form of Eq. (1), where is given by

 pcotδ(p)=−4πMU11√Mε0−√M(ε0+δ−E)√Mε0−√M(ε0+δ−E)+U212U11U22√M(ε0+δ−E). (49)

At low scattering energy and near the FR (), it can be well approximated as

 pcotδ(p)≃−4πMU11E−δE−δ+2U212U11U22ε0. (50)

Thus, the coupling constants are related to the physical observables through the following relations:

 U11=4πabgM,   U22=4πM1√Mε0,   γBΔ=2U212U11U22ε0. (51)

In terms of and , the effective range can be explicitly expressed as

 reff=−16π2M2U212(Mε0)3/2, (52)

which indicates that the effective range is always negative. From Fig. 1, we find that the binding energy equals the Zeeman energy splitting at the resonance Bruun (); i.e.,

 ε0=EZ(B=B0). (53)

### iii.2 Atom-molecule model as a low-energy effective theory

The phenomenological coupling in the atom-molecule model is related to physical observables as . From Eq. (51) we can identify . Therefore, can be expressed in terms of and as

 g=U12√(Mε0)3/22π. (54)

This expressions shows explicitly how the phenomenological coupling is related to the microscopic parameters. In the following we show that the atom-molecule model can be viewed as a low-energy effective theory in the limit while keeping the phenomenological coupling finite (hence, ). In this limit, we have , , and , which leads to . Therefore, the relations between and can be well approximated as

 (V11−V212V22)−1 = (U11−U212U22)−1−η(Λ) ≃ 1U11−η(Λ), (V22−V212V11)−1 = (U22−U212U11)−1−η(Λ) ≃ 1U22(1+U212U11U22)−η(Λ), = U12U11U22−U212≃U12U11U22. (55)

Comparing with the atom-molecule model, we identify

 u=U11,      u0=V11−V212V22. (56)

To arrive at the atom-molecule model we introduce an auxiliary field and integrate out the closed-channel fermions. Then the effective action can be expressed as , where

 Sf = ∫dx[∑σ=↑,↓ψ†σ(x)(∂τ−∇22M−μ)ψσ(x)+(V11−V212V22)ψ†↑(x)ψ†↓(x)ψ↓(x)ψ↑(x)], Sb = ∫dx⎡⎢⎣−|Φm(x)|2V22−Trln⎛⎜⎝−∂τ+∇22M+μ−12εthΦm(x)Φ†m(x)−∂τ−∇22M−μ+12εth⎞⎟⎠⎤⎥⎦, Sbf = ∫dxV12V22[Φm(x)ψ†↑(x)ψ†↓(x)+Φ†m(x)ψ↓(x)ψ↑(x)]. (57)

Here we have introduced the chemical potential and used to denote the open-channel fermions. Using the fact we find that the fermion part corresponds precisely to the atom part of the atom-molecule model.

Next we consider the molecule part and the atom-molecule coupling part . The inverse propagator for the boson field is given by

 D−1m(x,x′)=δ2Sb[Φ†m,Φm]δΦ†m(x)δΦm(x′). (58)

In the momentum space, it can be explicitly evaluated as

 D−1m(ω,q)=−1V22+∑k1ω+iϵ+2μ−12εq−εth−2εk. (59)

At low energy, i.e., , it can be expanded in terms of and . We have

 D−1m(ω,q)≃d0+d1(ω−q24M), (60)

where

 d0 = M4π√M(εth−2μ)−(1V22+MΛ2π2), d1 = M28π√M(εth−2μ). (61)

It becomes evident in the following that the low-energy expansion (59) corresponds to the leading-order expansion in . For large , we have . Therefore, and can be well approximated as

 d1≃α=M28π√Mε0 (62)

and

 d0 ≃ M4π√M(εth−2μ)−1U22 (63) −U212U11U222+V212V22(V11V22−V212) = M4π[√M(ε0+δ−2μ)−√Mε0] +(V12V22)21u0−(U12U22)21u ≃ α[δ−2μ+1αu0(V12V22)2−1αu(U12U22)2].

Then we define a normalized molecule field,

 ϕm(x)=√αΦm(x), (64)

which corresponds to the molecule field used in the atom-molecule model. The effective actions become

 Sb ≃ ∫dx ϕ†m(x)[∂τ−∇24M−2μ +δ+1αu0(V12V22)2−1αu(U12U22)2]ϕm(x), Sbf ≃ ∫dx1√αV12V22[ϕm(x)ψ†↑(x)ψ†↓(x)+H.c.]. (65)

Using the relations between and in Eq. (54) we obtain

 1√αV12V22=1√αU12U11U22u0=g1−η(Λ)u=g0 (66)

and

 δ+1αu0(V12V22)2−1αu(U12U22)2 (67) = δ+g20u0−g2u=δ+g2η(Λ)1−η(Λ)u=δ0.

Here we have used the definition of the atom-molecule coupling

 g=√γBΔu=√2U212U22ε0=U12√(Mε0)3/22π. (68)

Therefore, we have shown that the atom-molecule model is a low-energy effective theory of the two-band model in the limit (and hence ), while keeping the phenomenological atom-molecule coupling finite. In the atom-molecule model, the energy scale is hidden and is automatically sent to infinity.

We can also work out the next-to-leading-order low-energy expansion of the molecule part . It is quartic in and corresponds to the two-body interaction of the closed-channel bound states. We have

 SNLOb=124πam2M∫dx |ϕm(x)|4, (69)

where is the scattering length of the closed-channel molecules. In the limit , this contribution can be safely neglected. However, for realistic systems, is large but finite, this term may be important for the stability of polarized superfluidity Zhai ().

## Iv Resonant Fermi Gas as a Two-Band Superfluid

Starting from the two-channel Hamiltonian (35) and (36), we naturally have a two-band description of the superfluid state which is analogous to the BCS theory of two-band superconductors two-band (); two-band-iskin (). The molecule binding energy appears explicitly in this theory as the band offset. In the dilute limit , we expect that the two-band description recovers the atom-molecule model description.

### iv.1 Superfluid Phase

Following the standard field theoretical treatment, we introduce the auxiliary pairing fields

 Φ(x)=(Φ1(x)Φ2(x))=(V11V12V21V22)(φ1(x)φ2(x)), (70)

where , with being the imaginary time, apply the Hubbard-Stratonovich transformation, and integrate out the fermion fields. The partition function of the system can be expressed as

 Z=∫[dΦ†][dΦ]exp(−Seff). (71)

 Seff=−∫dx Φ†(x)V−1Φ(x)−∑n=1,2TrlnG−1n[Φn(x)], (72)

where the inverse fermion Green’s functions are given by

 (73)

Here we have defined and , with being the fermion chemical potential.

In the superfluid phase, the pairing fields have nonzero expectation values. We write

 Φn(x)=Δn+ϕn(x), (74)

where the constants and serve as the order parameters of superfluidity. Note that both and are superpositions of the pair potentials and . The order parameters and are both finite quantities, in contrast to the atom-molecule model. The effective action can be expanded in terms of the fluctuations . In the following, we evaluate the effective action up to the Gaussian fluctuations, i.e., . First, we consider the mean-field part . It can be evaluated as , where the grand potential is given by

 Ω0 = (75) − 2T∑n=1,2∑kln(1+e−Enk/T).

Here and the dispersions are defined as and . Note that we have used the renormalized coupling . The grand potential here is free from the cutoff for arbitrary values of and . Therefore, and are two independent physical quantities in the present two-band theory.

The contribution from Gaussian fluctuations is given by

 Sg=12∑Qϕ†(−Q)M(Q)ϕ(Q), (76)

where with ( integer) and . The inverse boson propagator is a